If you talk to true fasting purists, yes it can. In reality you can still get most of the benefits of fasting while still consuming some calories (250-400 for example). Even on a keto diet with a normal caloric intake you can get up to 70% of the health benefits because of the low level of mTOR and IGF-1 (according to Dr.Fung).
That having been said in the context of the primer 52 I still recommend th full fast. But having a sitck of gum once in a while should not make you lose the benefits of that day.
BTW, the bad breath while fasting is normally a good sign. It is often due to acetone being released by the breath and sweat ans indicates the production and use of ketone bodies.